How to verify authenticity of AirPods Pro 2nd Gen besides the serial number?

Not even the serial number will help in verifying if they are authentic. The only way to know for sure is to purchase them from Apple or Authorized Apple Seller. Any identification, including the serial number, is easily copied. The counterfeit market for Apple accessories is huge, so they would more than likely be fake if they were purchases at any third party location. You will find that fake AirPods may have these issues:

• Poor Audio Quality
• Find My may not work
• Noise Cancellation may not work
• Will not get AppleCare coverage
• May have problems pairing or keeping a connection

That may very well be an indicator that they are fake, but as long as you are happy with them and they are working correctly, then that is all that matters. Again, the only way to know for sure is to buy from Apple or Authorized Apple seller.

I don't have a pair of 2nd gen Airpod Pros to verify, but you are correct that normally you will see the model number/serial number just below the ear piece.